Autocorrelation Proof

Property 1: For any stationary process,  γ₀ ≥ |γ_i| for any i

Proof: For any stationary process y_i with mean µ, define z_i = y_i – µ. Then it is easy to see that z_i is a stationary process with mean zero. Also

(including the case where k = 0) which means that it is sufficient to prove the property in the case where the mean is zero.

Now suppose that E[z_i–cz_i+k] ≥ 0 for some real number c. Now for any real number c, it follows that

and so

If γ_k ≥ 0, let c = 1, while if γ_k < 0, let c = -1. Then